On the other hand, since \(\frac{a(p)}{\zeta }\in \mathbb {R}\) , then \(\beta _p=\overline{\alpha _p}\)  and by Deligne's theorem [1]} we have \(|\alpha _p|=|\beta _p|=1\) . Thus, we may write \(\alpha _p=e^{i\theta _p}\)  and \(\beta _p=e^{-i\theta _p}\)  for some \(\theta _p\in [0,\pi ]\) . Inserting this in (REF ) we obtain the desired identities.
